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I am applying for graduate school in pure mathematics and I recently got very interested in C*-algebra.

I am definitely wrong but I get the feeling that C*-algebras is not as popular as other areas of pure mathematics like number theory, analysis, algebraic geometry, etc. It also seems that most top ranked universities like MIT, Harvard, Stanford, Princeton, etc do not have any active research group in C*-algebras.

If my observations is right, then what is the reason? Is it because C*-algebra is harder than other areas of pure mathematics or is it because it is still a young area of pure mathematics?

Given that I am interested in C*-algebras and most top ranked universities are not active in this area, where can I apply? Also, will doing graduate work in C*-algebras instead other more popular areas of pure mathematics have a negative effect on my academic career?

Nina Wang
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    Both UCLA and UC Berkeley have strong groups working on operator algebras and related topics. And there is this practical site that allows you to search for operator algebraists: https://operatoralgebras.org/directory.html In general, I think your comparison with other fields is off - number theory or algebraic geometry are just much broader than $C^\ast$-algebras. – MaoWao Jan 03 '20 at 12:24
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    I would be interested to know if there is any truth in the OP's impression. As a quantum physicist I always thought C* algebras are an important part of modern math and even more so in the '30s- up until probably '50s-'60s. Am I wrong? – lcv Jan 03 '20 at 12:41
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    I think the OP's impression is indeed wrong. There are several groups worldwide working on operator algebras with a focus on C-algebras: in Germany for example there is a big group in Münster and there are groups in Göttingen and in Erlangen (the latter with a focus on representation theory and on topological insulators). In the UK there is Oxford, which recently hired a professor working in the classification programme of nuclear simple C-algebras, Glasgow has a big group in operator algebras. The Newton Institute at Cambridge had a whole programme on operator algebras in 2017. – Ulrich Pennig Jan 03 '20 at 13:21
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    The above is of course by no means meant to be a complete list. – Ulrich Pennig Jan 03 '20 at 13:25
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    @lcv: my impression is that the connection to physics has faded as a motivation (though there is certainly still good work being done in that direction). But C*-algebra has grown dramatically since the '60s, with Fields Medal level work being done establishing connections with other mathematical areas. – Nik Weaver Jan 03 '20 at 15:35
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    I think that the OP (understandably, we've all been there, we all still do it from time to time) errs by extrapolating or using an implicit universal quantifier when these things vary from country to country and generation to generation. There is also the need to remember these things are relative; Cstar algebras might not be as "popular" as X but they are a lot more popular than Y or Z – Yemon Choi Jan 03 '20 at 15:53
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    @UlrichPennig I think (but have not really tested this theory against the evidence) that because of the tenure system in North America the Elliott-Toms-Winter-fueled resurgence has been slower to translate into new hires than in Germany, Scotland or Wales (also waves to Xin at QMUL). She is correct to note that at MIT, Stanford, Harvard op alg is not a thing; while at Berkeley aren't they down to Voiculescu and Jones as emeritus? – Yemon Choi Jan 03 '20 at 15:56
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    @YemonChoi: "at Berkeley aren't they down to Voiculescu and Jones as emeritus" --- I hadn't realized that. Kind of depressing. – Nik Weaver Jan 03 '20 at 16:03
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    @NikWeaver Well I'd forgotten about Marc Rieffel, tbf, but yes time has done its thing... – Yemon Choi Jan 03 '20 at 16:07
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    @YemonChoi: But I don't think Marc is very active. I looked at the department website and did see a couple of postdocs in the area, though. – Nik Weaver Jan 03 '20 at 16:08
  • There's Vanderbilt since Jones went there. – AHusain Jan 03 '20 at 16:40
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    I'll throw in a plug for the University of Waterloo in Canada. I'm a master's student there right now in the pure math department, and there seems to be a lot of interest in $C^*$-algebras. I believe there is a course offered every other year on this topic. – Dave Jan 03 '20 at 23:42
  • In addition to the already mentioned schools, the University of Nebraska has at least 3 professors working in $C^*$-algebras, including a new hire: Chris Schaffhauser, who did a postdoc at Waterloo. UNL also has an active group of graduate students who hold learning seminars. – Eric Canton Jan 04 '20 at 05:05
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    @Dave TBF, with all due respect to Matt and Ken (Laurent is really more of an operator theorist than an operator algebraist) if we named every place that had some Cstar algebraists, this comment thread would triple in length and still not shed that much light on the (in)correctness of the OP's question, because these things are relative. – Yemon Choi Jan 04 '20 at 05:51
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    @EricCanton Allan D and David P are more in the vein of non-self-adjoint operator algebras rather than Cstar algebras per se. Chris S is the only out and out Cstar-algebraist, last time I checked. – Yemon Choi Jan 04 '20 at 05:54
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    One thing I haven't seen in the comments so far is that, when you're going to graduate school, you don't really know what you want to do, so don't necessarily fret too much about it. I went to Chicago thinking I wanted to study C* algebras, and I ended up in representation theory. Not a world apart, but not at all the same! – LSpice Apr 30 '20 at 02:04

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One way to tell how active a field is is by looking at what's appearing on the arXiv in that area. I think that will show you that operator algebra is a robust subject with a lot of activity.

In the comments, MaoWao points out that UC Berkeley and UCLA have very strong operator algebra groups, and Ulrich Pennig mentions groups in Münster, Göttingen, Erlangen, and Glasgow as places with substantial groups. Copenhagen is another good example.

On the other hand, the OP's observation that most top schools don't have an operator algebra group is quite correct. I would think that simply has to do with the size of the field --- there aren't enough C*-algebraists to populate that many departments. The two Fields Medals in the subject (to Connes and Jones) show that people in other areas do respect the field, I think.

The question is partly about career advice. All I can do there is report my impression that C*-algebraists don't seem to have more trouble finding employment than mathematicians of equal ability in other areas.

You ask which schools you should apply to --- in the comments MaoWao gave this site, which I was not aware of, which lists operator algebraists worldwide. It looks pretty complete to me.

However, my last comment is that as a prospective graduate student, you are at a very early stage to be settling on a specialty. Not saying you shouldn't, but I think most of us would recommend putting your main effort into getting a broad education during the first two years of grad school.

Related MathOverflow questions:

What are the applications of operator algebras to other areas?
States in C*-algebras and their origin in physics?
Quantum functional analysis
applications of C*-algebras in the field of PDEs

jeq
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Nik Weaver
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  • Oh, one other thing. You ask which schools you should apply to --- if you're in the US, Berkeley and UCLA for sure, those are the top ranked schools, but there are plenty of good places with more than one operator algebraist. I don't know of any sort of directory that would tell you where, but just looking at any particular department's website should tell you in most cases whether they fit the bill. – Nik Weaver Jan 03 '20 at 14:55
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    Regarding a directory of operator algebraists, there is the one I linked in a comment above: operatoralgebras.org/directory.html But I only stumbled across it by chance, so I don't know how complete it is. Do you happen to know (at least you are included)? – MaoWao Jan 03 '20 at 15:11
  • @MaoWao: oh, I didn't see that! I'll add this to my answer. – Nik Weaver Jan 03 '20 at 15:15